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 A103519 a(1) = 1, a(n) = sum of n successive numbers starting with a(n-1) + 1. 2
 1, 5, 21, 94, 485, 2931, 20545, 164396, 1479609, 14796145, 162757661, 1953092010, 25390196221, 355462747199, 5331941208105, 85311059329816, 1450288008607025, 26105184154926621, 495998498943605989, 9919969978872119990 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, the triangular series (1, 3, 6, ...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016 LINKS Harvey P. Dale, Table of n, a(n) for n = 1..449 FORMULA a(n+1) = k(k+1)/2 - a(n)*(a(n)+1)/2, where k = a(n) + n +1. a(n) = Sum_{i=0..n} (n!/(n-i)!) * (n-i)(n-i+1)/2 = Sum_{i=0..n} (n!/(n-i)!) * A000217(n-i). For n > 2, a(n) = (3*n*(n-1)/2)*floor((n-2)!*e) + n, where e=exp(1). - Max Alekseyev, Feb 14 2005 a(n) = n*a(n-1) + n*(n+1)/2. - Emeric Deutsch, Mar 16 2008 a(n) ~ 3*sqrt(Pi/2)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016 EXAMPLE a(2) = 2 + 3 = 5, a(3) = 6 + 7 + 8 = 21, a(4) = 22 + 23 + 24 + 25. MAPLE a[1]:=1: for n from 2 to 20 do a[n]:=n*a[n-1]+(1/2)*n*(n+1) end do: seq(a[n], n=1..20); # Emeric Deutsch, Mar 16 2008 MATHEMATICA RecurrenceTable[{a[1]==1, a[n]==n*a[n-1]+(n(n+1))/2}, a, {n, 20}] (* Harvey P. Dale, Nov 05 2013 *) PROG (PARI) { t(n) = n*(n+1)/2 } { a(n) = sum(i=0, n, n!/(n-i)!*t(n-i)) } { a2(n) = 3*t(n-1)*floor((n-2)!*exp(1))+n } \\ Max Alekseyev, Feb 14 2005 CROSSREFS Cf. A103520. Sequence in context: A116904 A126952 A273570 * A178876 A202513 A343349 Adjacent sequences: A103516 A103517 A103518 * A103520 A103521 A103522 KEYWORD easy,nonn AUTHOR Amarnath Murthy, Feb 10 2005 EXTENSIONS More terms from Max Alekseyev, Feb 14 2005 STATUS approved

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Last modified December 2 17:17 EST 2022. Contains 358510 sequences. (Running on oeis4.)